Does watch(provider) update children of parent widget? - flutter

Question

I'm trying to update both Pages with one UserInteraction, therefore trying to access the same Stream in both Pages, with the Riverpod library.
Now to explain it further. When I pass the Stream to the CustomerPage I'm able to get the data (the String Anton). and when I click on the Button that triggers the change in FireStore, the String gets updated to "Marco" in the ParentWidget, when I go back to it. But it doesn't change in the CustomerPage unless I reopen the Page via the RaisedButton in the ParentWidget.
But I want it to update after I click the Button on the CustomerPage.
I hope this makes it clearer.
class ParentWidget extends ConsumerWidget{
Widget build(BuildContext context, ScopedReader watch){
Stream<DocumentSnapshot> doc = watch(streamProvider.stream);
return Container(
child: Column(
children: [
Text(doc.name), //Lets say the name is Anton,
RaisedButton(
child: Text(" road to CustomerPage"),
onPressed:(){
Navigator.of(context).pushNamed(RouteGenerator.customerPage, arguments: doc);
},), //RaisedButton
],), //Column
); //Container
}
}
class CustomerPage extends StatelessWidget{
Stream<DocumentSnapshot> docStream
CustomerPage({this.docStream});
Widget build(BuildContext context){
return Column(
children: [
Text(docStream.name) //Here is also Anton
RaisedButton(
child: Text("Change Name"),
onPressed: () {
context.read(streamProvider).changeName("Marco");
},), //RaisedButton
]
); //Column
}
}

Answer 1

On how I've understood so far is that, riverpod allows you to fetch the state of a provider, which basically is a value(?), that's why it's sufficient to just watch it in any Widget you want to access it's data from. There is no need anymore (just speaking for my case), to let Widgets pass the around in the App.
Down below is the solution which i believe to be right.
It also doesn't matter on how many times I call the provider. It's always going to be same Instance. For my case it means, that doc and doc2 are the same.
I hope this makes it clearer.
class ParentWidget extends ConsumerWidget{
Widget build(BuildContext context, ScopedReader watch){
Stream<DocumentSnapshot> doc = watch(streamProvider.stream);
return Container(
child: Column(
children: [
Text(doc.name), //Lets say the name is Anton,
RaisedButton(
child: Text(" road to CustomerPage"),
onPressed:(){
Navigator.of(context).pushNamed(RouteGenerator.customerPage);
},), //RaisedButton
],), //Column
); //Container
}
}
class CustomerPage extends ConsumerWidget{
Widget build(BuildContext context, ScopedReader watch){
Stream<DocumentSnapshot> doc2 = watch(streamProvider.stream);
return Column(
children: [
Text(doc2.name) //Here is also Anton
RaisedButton(
child: Text("Change Name"),
onPressed: () {
context.read(streamProvider).changeName("Marco");
},), //RaisedButton
]
); //Column
}
}